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I have been working on a simulator for bending of beams and came now to a tricky doubt: What should be the difference between a linear and non linear solution in this case (graphic at bottom)?

The solution of the following ODE gives us the non-linear curvature:

enter image description here

and for very small angles (dy/dx)^2 will tend to zero, so we can linearize (Sorry, $dy=dv$ in this image):

enter image description here

So we integrate the following equation (I used the bvp4c function on Matlab) , that includes the curvature, to obtain the deflection of the beam:

enter image description here

In red is the non-linear solution and in Blue the linear solution.

enter image description here

My doubt is: at the middle of the $x$-axis $dy/dx=0$ in both curves, should I expect then also to have the same value of $y$, since at that precise point I could also cancel $dy/dx$ in the formula and both equations would look the same? In other words, what should I expect as a difference between a linear and a non-linear solution in such equations?

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I don't think that the deflection of the beam in the centre should be the same. You use two different relations to compute curvature from the deflection. Since, you use the same load, the two different relation have to result in different solutions. In general it's hard if not impossible to say in advance what the differences in the solution will be. This strongly depends on the simplifications you make on the way to the linear model.

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Even though dy/dx cancels out - in both cases - at the centre, this does not mean automatically, that y will also be the same. –  Dohn Joe Apr 22 at 10:24
    
Thanks very much Dohn. It really helped me. –  user7991 Apr 22 at 12:15

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